32.5 Matrices and Transformations

The most important use of matrices lies in representing linear transformations
on a vector space.

How?

A matrix represents the tranformation which takes
the first basis vector
into first column of the matrix, second basis vector into the second column
of the matrix, j-th basis vector into j-th column.

What does it do to other vectors?

Remember that any other vector, say
v&LongRightArrow;
can be expressed as
a linear combination of the basis vectors:v&LongRightArrow;
gets transformed by the transformation to
that same linear combination of the column vectors of the matrix.

For example, the sum of the first two basis vectors gets mapped into the sum of the first two columns of the matrix; the average of the two basis vectors into the average of the columns, and so on.

Notice that the same transformation acting on vectors will usually be described
by a different matrix if you use a different basis.